Extending functors on the category of manifolds and submersions


Let $G$$G$ be a compact Lie group. Consider the corresponding functors, where all manifolds are endowed with a smooth $G$$G$ action and all maps are equivariant. Again, cobordism classes form a group, but in this case the transversality theorem does not hold and so the question comes up how to construct induced maps. A similar problem with transversality occurs for maps between infinite dimensional manifolds.

In this note we explain how to overcome this difficulty by extending functors with certain properties, defined on the category of smooth $G$$G$-manifolds and equivariant submersions, to functors, where induced maps are defined for all smooth equivariant maps.

1 Introduction

Let $G$$G$ be a compact Lie group. Given a contravariant homotopy functor $h_G(M)$$h_G(M)$ from the category of smooth $G$$G$-manifolds (either finite dimensional or Hilbert manifolds) and morphisms which are equivariant submersions to the category of abelian groups we would like to construct an extension of induced maps for arbitrary equivariant smooth maps. By a homotopy functor we mean that if we have a homotopy through submersions, then the induced maps agree. We also want the extended functor to be a homotopy functor on all smooth maps.

The following is a necessary condition:

Condition: If $f: E \to M$$f: E \to M$ is a smooth equivariant vector bundle then $f^*$$f^*$ is an isomorphism.

Theorem 1.1. Let $h_G(M)$$h_G(M)$ be a contravariant homotopy functor from the category of smooth $G$$G$-manifolds (either finite dimensional or Hilbert manifolds) and morphisms which are equivariant submersions, to the category of abelian groups such that the condition above is fulfilled. Then there is a unique extension to a homotopy functor on all equivariant smooth maps. If $h_G(M)$$h_G(M)$ is a graded abelian group together with a natural Meyer-Vietoris sequence for every equivariant open covering $M = U \cup V$$M = U \cup V$ (a cohomology theory on the category of smooth manifolds and submersions), then the extension to the category of smooth manifolds and all smooth maps is again a cohomology theory: The Meyer-Vietoris sequence is again natural.

2 The construction

The proof is given by factoring an arbitrary smooth map through the composition of an embedding and a submersion to construct the extension in a unique way. The main thing left to be shown is functoriality. We prepare the construction of the extension with considerations concerning equivariant tubular neighborhoods.

Lemma 2.1. Every equivariant closed smooth embedding has an equivariant tubular neighborhood, which is unique up to homotopy through submersions.

Proof. Tubular neighborhoods can be constructed as in the finite dimensional setting. This is carried out in [Lang1999] using the concept of sprays. Instead one can use a Riemannian metric. From both data one gets the exponential map, which is a diffeomorphism on a small neighborhood of the zero section of the normal bundle to its image. Then one uses partition of unity to construct a diffeomorphism by shrinking the normal bundle to the open neighborhood. All this works equivariantly for compact Lie group actions. The only difference is that one considers an equivariant Riemannian metric which one obtains by averaging a non-equivariant Riemannian metric. Then the exponential map is automatically equivariant. Similarly, the shrinking construction works equivariantly by using an equivariant partition of unity, which again can be obtained from a non-equivariant partition of unity by averaging. The uniqueness is obtained by using a relative equivariant tubular neighborhood

$\square$$\square$

Given an equivariant closed smooth embedding $g:P \to L$$g:P \to L$ with an equivariant tubular neighborhood $\psi:\nu \to L$$\psi:\nu \to L$ and projection map $p_\nu: \nu \to P$$p_\nu: \nu \to P$, denote $(p_\nu^*)^{-1} \psi^*$$(p_\nu^*)^{-1} \psi^*$ by $\widehat{g}$$\widehat{g}$. Since equivariant tubular neighborhoods are unique up to homotopy through submersions $\widehat{g}$$\widehat{g}$ is independent of this choice.

With this we define induced maps for an arbitrary equivariant smooth map $f: M \to N$$f: M \to N$ as follows. First, we factor $f$$f$ as the composition of the closed embedding $(\id,f): M \to M\times N$$(\id,f): M \to M\times N$ and the projection $p: M\times N \to N$$p: M\times N \to N$. Then we define

$\displaystyle f^*:=\widehat{(\id,f)} p^*: h_G(N) \to h_G(M).$

We have to show:

1) $h_G$$h_G$ coincides with the previously defined functor for equivariant submersions.

2) $h_G$$h_G$ is a homotopy functor from the category of smooth (Hilbert) G-manifolds and equivariant smooth maps.

Proposition 2.2. The definition of induced maps coincides with the previous definition for submersions.

Proof. Let $f: M \to N$$f: M \to N$ be an equivariant submersion. We need to show that $p_\nu^*f^* = \psi^* p^*$$p_\nu^*f^* = \psi^* p^*$, which is equivalent to showing that $(f p_\nu)^*= (p\psi)^*$$(f p_\nu)^*= (p\psi)^*$ by functoriality for submersions. This holds since $f p_\nu$$f p_\nu$ is homotopic to $p\psi$$p\psi$ via submersions. A homotopy is given by mapping $(v,t)$$(v,t)$ to $p\psi(\rho(t)v)$$p\psi(\rho(t)v)$, where $\rho : [0,1] \to [0,1]$$\rho : [0,1] \to [0,1]$ is a smooth map, which is $0$$0$ near $0$$0$ and $1$$1$ near $1$$1$. It is a homotopy via submersions, since for $\rho(t )\ne 0$$\rho(t )\ne 0$ one uses that $p\psi$$p\psi$ is a submersion and for $\rho(t)=0$$\rho(t)=0$ one uses that $f p_\nu$$f p_\nu$ is a submersion.

$\square$$\square$

It is clear from the construction that this extension is unique. The fact that it depends only on the homotopy class of the map follows from the fact that if $f$$f$ is homotopic to $f'$$f'$ then $(\id,f)$$(\id,f)$ is isotopic to $(\id,f')$$(\id,f')$.

We are left to show functoriality. This is done in few steps.

Lemma 2.3. Given a pullback square

$\displaystyle \xymatrix{ M \ar[d]^{p} \ar[r]^{f} & N\ar[d]^{p'} \\ M'\ar[r]^{f'} & N',\\}$

where $f'$$f'$ is an equivariant closed smooth embedding and $p'$$p'$ is an equivariant submersion. Then $p^*\widehat{f'} =\widehat{f} p'^*$$p^*\widehat{f'} =\widehat{f} p'^*$.

Proof. Since this is a pullback square, $f$$f$ is an equivariant closed smooth embedding and $p$$p$ is an equivariant submersion, therefore, all induced maps are defined.

In Theorem 6.7 in [Hirsch1976] the following was proven for non-equivariant maps. In the diagram above, if $M'$$M'$ is compact, then suppose given tubular neighborhoods $U\subseteq N$$U\subseteq N$ of $M$$M$ and $E \subseteq N'$$E \subseteq N'$ of $M'$$M'$. Let $D \subseteq U$$D \subseteq U$ be a disk bundle such that $p'(D)\subseteq E$$p'(D)\subseteq E$, then one can approximate $p'$$p'$ by $q'$$q'$, via a homotopy $h_t$$h_t$, such that $q'|_D$$q'|_D$ is a vector bundle map, $h_t=p'$$h_t=p'$ on $M\cup (N \setminus U)$$M\cup (N \setminus U)$ for $0\leq t\leq 1$$0\leq t\leq 1$ and $h_t^{-1}(N'\setminus M')=N\setminus M$$h_t^{-1}(N'\setminus M')=N\setminus M$ for $0\leq t\leq 1$$0\leq t\leq 1$.

The same argument works for non-compact $M'$$M'$ if one replaces $D$$D$ by a subbundle $D_\varepsilon=\{(x,v)\in U|\: ||v|| \leq \varepsilon(x)\}$$D_\varepsilon=\{(x,v)\in U|\: ||v|| \leq \varepsilon(x)\}$, for some map $\varepsilon:M \to \mathbb{R}_{>0}$$\varepsilon:M \to \mathbb{R}_{>0}$. In order to have a disk bundle as in Hirsch's Theorem, one has to renormalize the metric. In addition, by taking equivariant tubular neighbourhoods and an equivariant map $\varepsilon$$\varepsilon$ (which can be constructed, since $G$$G$ is compact), one obtains the same result in the equivariant setting. Notice, that since the subspace of all submersions is open in the space of all maps, one can choose $h_t$$h_t$ such that it a submersion for all $t$$t$. Since all induced maps remain unchanged along the homotopy we may assume with no loss of generality that $p'=q'$$p'=q'$. Hence, we obtain the following commutative diagram where all maps are submersions:

$\displaystyle \xymatrix{ M \ar@{->>}[d]^{p} & \nu \ar@{->>}[d]^{p''} \ar[r]^{\psi} \ar@{->>}[l]_{p_\nu}& N \ar@{->>}[d]^{p'} \\ M' &\nu' \ar[r]^{\psi '} \ar@{->>}[l]_{p_\nu '}& N' \\}$

This proves the lemma since the induced maps for submersions are functorial.

$\square$$\square$

Lemma 2.4. Let $f: M \to N$$f: M \to N$ and $g: N \to P$$g: N \to P$ be equivariant closed smooth embeddings, then $\widehat{gf}=\widehat{f}\widehat{g}$$\widehat{gf}=\widehat{f}\widehat{g}$.

Proof. Denote by $\nu (M,N), \nu (N,P)$$\nu (M,N), \nu (N,P)$ and $\nu (M,P)$$\nu (M,P)$ the normal bundles of the image of $f,g$$f,g$ and $gf$$gf$ respectively. We have the following commutative diagram

$\displaystyle \xymatrix{ & \nu_{(M,P)}\ar[d]^{\exists \cong} \ar[dr] \ar[drr] \ar[ddl]&& \\ &U\ar[r] \ar[d]&\nu_{(N,P)} \ar[r] \ar[d]&P\\ M&\nu_{(M,N)}\ar[r] \ar[l]& N&\\}$

where $U$$U$ is the pullback of the square. Since all maps are submersions the result follows from the commutativity and the naturality for induced maps for submersions.

$\square$$\square$

Lemma 2.5. If $f: M \to N$$f: M \to N$ is an equivariant closed smooth embedding then $f^*=\widehat{f}$$f^*=\widehat{f}$.

Proof. Look at the following commutative diagram:

$\displaystyle \xymatrix{ M \ar@{^{(}->}[r]^{(\id,\id)} \ar[dr]^{\id} & M\times M \ar[d]^{p} \ar@{^{(}->}[r]^{(\id,f)} & M\times N \ar[d]^{p} \\ & M \ar@{^{(}->}[r]^{f} & N \\}$

The right square is a pullback, so we can use Lemma 2.3. The triangle on the left is used to compute the induced map for the identity on $M$$M$, so we can use Proposition 2.2. Then we use Lemma 2.4, which shows that the map induced by the upper row is equal to the map induced by $(\id,f)$$(\id,f)$. Putting this all together we get:

$\displaystyle f^*=\widehat{(\id,f)}p^*=\widehat{(\id,\id)}\widehat{(\id,f)}p^*= \widehat{(\id,\id)}p^*\widehat{f}=\id^*\widehat{f}=\widehat{f}$
$\square$$\square$

Remark 2.6. What we saw so far implies that $f^*=(\id,f)^*p^*$$f^*=(\id,f)^*p^*$, and that for equivariant closed smooth embeddings $f: M \to N$$f: M \to N$ and $g: N \to P$$g: N \to P$ we have $(gf)^*=f^*g^*.$$(gf)^*=f^*g^*.$

We now prove functoriality in the general case:

Proposition 2.7. For equivariant smooth maps $f: M \to N$$f: M \to N$ and $g: N \to P$$g: N \to P$ we have

$\displaystyle (gf)^*=f^*g^*.$

Proof. We prove the statement by showing that certain induced maps commute. For this we use the following commutative diagram

$\displaystyle \xymatrix{ & & & M\times P \ar@{->>}[dddl]^{p_P} \ar@{^{(}->}[ld]_{(id,f,id)} \\ M \ar@{^{(}->}[r]_{(id,f)} \ar@{^{(}->}[rrru]^{(id,gf)} & M\times N \ar@{^{(}->}[r]_{(id,id,g)} \ar@{->>}[d]^{p_N} & M\times N \times P \ar@{->>}[d] ^{p_{N\times P}}\\ & N \ar@{^{(}->}[r]_{(id,g)} & N\times P \ar@{->>}[d] ^{p_P}\\ & & P\\ }$

The proof consists of three parts.

1) The middle square is a pullback, so by Lemma 2.3 we have $p _{N}^*(\id,g)^*=(\id,\id,g)^*p _{N\times P}^*$$p _{N}^*(\id,g)^*=(\id,\id,g)^*p _{N\times P}^*$.

2) Lemma 2.5 for the upper triangle implies:

$\displaystyle (\id,f)^*(\id,\id,g)^*=(\id,gf)^*(\id,f,\id)^*.$

3) For the right triangle, we want to show that

$\displaystyle p _{P}^*=(\id,f,\id)^*p _{P}^*.$

To see that, choose a of tubular neighborhood $\nu$$\nu$ of $M$$M$ in $M\times N$$M\times N$ and take $\nu\times P$$\nu\times P$ to be a tubular neighborhood of $M\times P$$M\times P$ in $M\times N\times P$$M\times N\times P$. Then the following diagram commutes

$\displaystyle \xymatrix{ M\times P \ar@{->>}[dr]^{p_P} &\nu \times P \ar@{^-->>}[l]_{(p_{\nu},\id)} \ar[r]^{(\psi,\id)} & M\times N \times P \ar@{^-->>}[dl]_{p_P} \\ & P & \\}$

Since all the maps here are submersions this implies what we wanted.

All this implies that:

$\displaystyle f^*g^*=(\id,f)^*p _{N}^*(\id,g)^*p _{P}^*= (\id,f)^*(\id,\id,g)^*p _{N\times P}^*p _{P}^*= \\ =(\id,f,fg)^*p _{P}^* =(\id,gf)^*(\id,f,\id)^* p _{P}^*=(\id,gf)^*p _{P}^*=gf^*$
$\square$$\square$

3 Extensions of cohomology theories

Often the functors map to the category of graded groups and fulfil the axioms of a cohomology theory on the category of smooth manifolds and submersions. This means that it is a homotopy functor together with a Meyer-Vietoris sequence for each equivariant open covering $M = U \cup V$$M = U \cup V$ which is natural with respect to induced maps, meaning that if $g: M' \to M$$g: M' \to M$ is a submersion and $M' = U'\cup V'$$M' = U'\cup V'$ and $g(U') \subseteq U$$g(U') \subseteq U$ and $g(V')\subseteq V$$g(V')\subseteq V$, then the corresponding diagram of Meyer-Vietoris sequences and restriction of the induces maps commutes.

Lemma 3.1. In such a situation the extension of the functor to the category of smooth manifolds and smooth maps is again a cohomology theory.

Proof. We have to show that the Mayer-Vietoris sequence is natural with respect to all equivariant smooth maps of triads. We already know that it is natural with respect to equivariant submersions of triads. Another case where naturality follows from naturality for submersions is the following. Let $\pi:E\to B$$\pi:E\to B$ be an equivariant vector bundle, and $U,V$$U,V$ an equivariant open covering of $B$$B$ and denote by $U’=\pi^{-1}(U)$$U’=\pi^{-1}(U)$ and $V’=\pi^{-1}(V)$$V’=\pi^{-1}(V)$. Then the zero section $c:B\to E$$c:B\to E$ induces a map of triads (we call such a map a zero section of triads), where all induced maps are the inverses of the induced maps for the projection $\pi$$\pi$, hence the naturality follows from the naturality for submersions. Therefore, we will be done if we show that every equivariant smooth map of triads factors as the composition of a zero section of triads and an equivariant submersion of triads. Since we already know that every equivariant smooth map factors as the composition of a zero section of an equivariant vector bundle and a submersion, it will be enough to show that every zero section in an equivariant vector bundle can be factored as a zero section of triads and an equivariant submersion of triads. We use the notation as above, and let $U’’,V’’$$U’’,V’’$ be an open cover of $E$$E$. There exists an invariant Riemannian metric on $E$$E$ such that the intersection of $U’'$$U’'$ and $V’’$$V’’$ with the open unit disk bundle $D\subseteq E$$D\subseteq E$ contains the intersection of $U’$$U’$ and $V’$$V’$ with $D$$D$. Then the map $c:B\to E$$c:B\to E$ factors as a zero section of triads $B\to D$$B\to D$ (with the open covering $U’\cap D$$U’\cap D$ and $V’\cap D$$V’\cap D$) and the submersion (open inclusion) $D\to E$$D\to E$.

$\square$$\square$

4 Applications

This extension is used in our geometric description of ordinary cohomology of Hilbert manifolds [Kreck&Tene2016].

As another application we consider the functor assigning to a finite dimensional $G$$G$-manifold $M$$M$ the group $\mathcal N^k_G(M)$$\mathcal N^k_G(M)$ of cobordism classes of pairs $(N,f)$$(N,f)$, where $N$$N$ is a $G$$G$-manifold with $\dim(M)-\dim(N)=k$$\dim(M)-\dim(N)=k$ and $f:N\to M$$f:N\to M$ is a proper, equivariant, smooth map. This is a homotopy functor on the category of smooth $G$$G$-manifolds and equivariant submersions, where induced maps are given by the pullback. Our condition is in general not fulfilled. To achieve this one passes to the cofiltered limit $\overline {\mathcal N}^k_G(M) := \lim_E(\mathcal N^k(E)$$\overline {\mathcal N}^k_G(M) := \lim_E(\mathcal N^k(E)$, where the limit is taken over the cofiltered category of $G$$G$-vector bundles $E \to M$$E \to M$ with morphisms equivariant linear submersions $F \to E$$F \to E$. This way we make sure that our condition is fulfilled for $\overline{\mathcal N}^k_G(M)$$\overline{\mathcal N}^k_G(M)$. By standard considerations one proves that this is a generalized equivariant cohomology theory. When $G$$G$ is the trivial group this theory agrees with non-oriented cobordism.

If one replaces in this example $(N,f)$$(N,f)$ by $\mathcal (S,f)$$\mathcal (S,f)$, where $S$$S$ is a stratifold (see [Kreck2010]), one obtains another cohomology theory on the category of smooth $G$$G$-manifolds and equivariant maps, denoted by $\overline {\mathcal SH}^*_G(\_;\mathbb Z /2)$$\overline {\mathcal SH}^*_G(\_;\mathbb Z /2)$.

Remark 4.1. When $G$$G$ is finite one can show that $\overline {\mathcal SH}^*_G(\_;\mathbb Z /2)$$\overline {\mathcal SH}^*_G(\_;\mathbb Z /2)$ is naturally isomorphic to the cohomology of the Borel construction. We plan to study such extensions in a separate paper.